Chain Rule Question Using the equation below, use the chain rule on r(t)= to show the speed is given by S(t)=u'(t)√(f' (u)^2+g' (u)^2 ) I know how the chain rule works: f'[g(x)]*g'(x) I understand if you take the derivative of both you would get f'(u(t))*u'(t) and same for the g portion g'(u(t))*u'(t) Not sure how that applies to the speed, the formula they give looks like u'(t) has been factored out and then a distance formula. @FibonacciChick666 @dumbcow

Question

Chain Rule Question

Using the equation below, use the chain rule on r(t)=<f(u(t)),g(u(t))>

to show the speed is given by S(t)=u'(t)√(f' (u)^2+g' (u)^2 )

I know how the chain rule works:
f'[g(x)]*g'(x)

I understand if you take the derivative of both you would get f'(u(t))*u'(t) and same for the g portion g'(u(t))*u'(t)

Not sure how that applies to the speed, the formula they give looks like u'(t) has been factored out and then a distance formula.

@FibonacciChick666 @dumbcow

anonymous · Answer

To make it easier to read the starting equation is: $$r(t) = <f(u(t)),g(u(t))>$$

The equation we're trying to get to is:
$$S(t) = u'(t)\sqrt{f'(u)^{2}+g'(u)^{2}}$$

primeralph · Answer

The major thing here is arc length. Ever heard of that?

primeralph · Answer

@Charollete

primeralph · Answer

@sarahusher  Can you finish this?

anonymous · Answer

Yeah. I was going to mention that in my main question, it looks like arc length which is the integral of that I believe.

primeralph · Answer

Something like that. Apply the arc length definition and you should get the answer.

anonymous · Answer

Ahh, so instead of working forward, work backwards from the speed by integrating.

primeralph · Answer

Not really working backwards. Hold on.

fibonaccichick666 · Answer

r(t) is a position function correct?

anonymous · Answer

Yes I believe so

primeralph · Answer

In my opinion, S is usually arc length, which would make sense here.
V is speed mostly.

anonymous · Answer

well the question says Use the Chain Rule to show that the speed is given by $$S(t) = u'(t)\sqrt{f'(u)^{2}+g'(u)^{2}}$$

anonymous · Answer

but it looks similar to arc length.

primeralph · Answer

@mathstudent55 
I'm tired. Please handle this.

anonymous · Answer

S is primarily used to denote arc length, I have never seen it to denote speed...
is it possible that there is just a typo in the question?

primeralph · Answer

Or we're all just too old to remember basic calculus.

anonymous · Answer

The question says use the Chain Rule to show that the speed is given by the 2nd equation above.

primeralph · Answer

@sarahusher  Is very smart. She'll help you.
I'm going to bed.

anonymous · Answer

I'm with you guys though it does look as if its arc length. but the given function is for a trajectory, so I'm assuming you can basically show steps on how to get to that second equation, and with it then find the speed? 

Thanks @primeralph  have a good night.

primeralph · Answer

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